A New Bound on Odd Multicrossing Numbers of Knots and Links

An n-crossing projection of a link L is onto plane such that n points on are superimposed top each other at every crossing. We prove for all k?? and links L, the inequality c2k+1(L)?2g(L)+r(L)?1k2 holds, where c2k+1(L), g(L) r(L) (2k+1)-crossing number, 3-genus, number components respectively. This result used to new bound odd crossing numbers torus knots generalizes Jablonowski (see [M. Jab?onowski, Triple-crossing genus knot or knots, Topology Appl.285 (2020) 107389]). also upper 5-crossing 2-torus links. Furthermore, we improve lower bounds 82 with 2-crossing most 12. Finally, 7-crossing 5